Pressure at the Interface of Inner Conductor and Dielectric of Armorless Ocean Cable By P. G. BHUTA (Manuscript received February 8, 1960) Pressure at the interface of the inner conductor and the dielectric of the armorless ocean cable is experimentally determined for -plane stress boundary conditions. Polyethylene older and steel inner concentric cylinders are con- sidered as a mechanical model of the ocean cable. Equations arc derived, to give the pressure at the interface for plane stress and plane strain conditions. It is found that the theoretical values of the pressure calculated from equa- tions of elasticity are in good agreem.ent mith experimental values, even when one of the materials is polyethylene, which has a nonlinear stress-strain relationship in simple tension or compression. From previously measured values of bulk and Young's moduli, it is shown that, for an elastic analysis, Poisson's ratio for polyethylene is very nearly 0.5. An experimental verifica- tion of the plane stress solution given in this paper confi.rms the foregoing value. I. INTRODUCTIOX In the process of designing new armorless coaxial ocean cables it is important to know the pressure at the interface of the inner conductor and the polyethylene dielectric. Dotermuiation of pressure at the inter- face would enable one to study the decay of tension, hi a cable-laying machine, from the strength member to the holding surface, and to in- vestigate the pressure effects on the resistance and capacitance of the cable. A suitable mechanical model of the armorless ocean cable is a poly- ethylene outer cylinder concentric with a steel inner cylinder. The caljle is actually subjected, m three dimensions, to a pressure loading that vari'-s with the depth of the ocean bottom. Ilowever, for a section of the cable on a relatively flat ocean l)ottom one may assume that the cable is subjected to a uniform external pressure. 963 964 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 One may obtain an expression for the pressure at the interface from a two-dimensional model (i.e., for plane stress or plane strain) using the theory of elasticity. However, the result is questionable, since poly- ethylene has a nonlinear stress-strain relationship in simple tension or compression. Moreover the value of Poisson's ratio for polyethylene, fip , ranges from 0.3 to 0.5, and it is not possible to determine iXp accu- rately by a simple experiment. The value of the pressure at the interface, as obtained from such an equation, depends considerably upon the particular value of fip that is used. Hence, an experimental determination of the pressure at the interface is necessary, to investigate whether an elastic analysis is valid when one of the materials is polyethylene and to establish the value of Poisson's ratio for polyethylene. The pressure at the interface of the model is determined experimentally for plane stress boundary conditions. Equations are derived, based on the two-dimensional theory of elasticity, to give the pressure at the in- terface of the model for plane stress and plane strain conditions. It is found that the theoretical values of the pressure calculated from equa- tions of elasticity are in good agreement with experimental vahies. From previously measured values of bulk and Young's moduli, it is shown that, for an elastic analysis, Poisson's ratio for polyethylene is very nearly 0.5. For this reason and since the area of polyethylene is much greater than that of steel, an extension of the two-dimensional plane strain solution to a three-dimensional case of uniform pressure by the method of super- position would give the pressure at the inner conductor and dielectric interface to be equal, or at most slightly greater, than the external pres- sure. II. EXPERIMENTAL MODEL AND PROCEDURE Initially, the experunental model consisted of a cold-drawn steel tub- ing 30 inches long, 1.254 inches in outer diameter and 0.083 inch in wall thickness. Two SR-4 strain gages of type A7 were cemented on a O.OIO- inch-thick steel ring, which fitted snugly inside the steel tubing shown in Fig. 1. The steel ring containing the gages to which long leader wires were soldered was cemented in the tubing so that the it was at the same distance from either end. The gages were far enough from the ends to eliminate end effects. The tubing was subjected to an external hydrostatic pressure in in- crements of 500 psi up to 4500 psi when a maximum strain of approxi- mately 1000 microinches per inch, which is well within the elastic limit for the tubing, was recorded. The pressure was reduced to zero, and upon increasing the pressure again, the same calibration curve was obtained. I-RESSURE AT INTERFACE IX ARMORLESS OCEAN CABLE OHo ■30 i.a54 .it" SEE DETAIL A — GAGE 1 GAGE 2 _ "-^m tZZZZZZ222ZZZZl 0.083 1 " -^ 2 [^ 0.010' Fig. 1 — Steel tubing, showing location of strain gages. The experimental arrangement for calibration of gages is sho^^ni in Fig. 2. After calibration of gages, the polyethylene outer cylmder, whose in- ner diameter was 0.004 inch smaller than the outer diameter of the steel cylinder, was forced onto the steel cylinder in a universal testing ma- chine. The assembled model appeared as in Fig. 3. Before the model was subjected to external hydrostatic pressure, the strain gages were bal- anced to indicate zero strain by adjusting resistances in a multichannel switching-and-balancing unit. The experimental arrangement for the model is given in Fig. 4. It is not the intent of the present paper to evalu- ate the pressure produced at the interface by the interference fit. External hydrostatic pressure was applied in increments of 500 psi up to 2500 psi and corresponding strains were observed. The pressure was reduced to zero and the reproducibility of the data was checked. ^ ^"O" RING I / SEAL GAGE l--^--GAGE 2 ITTTTTTmTTTni ^^m^/^^^^^^^Z^^^^ '-STEEL Fig. 2 — Experimental arrangement for calibration. 9r»6 THE BKLL SYSTEM TECHNICAL JOURNAL, JULY 1960 Fig. 3 — Experimental model. -POLYETHYLENE Fig. 4 — Experimental arrangement for the model. III. EXPERIMENTAL RESULTS The calibration curves for external hydrostatic pressure versus the strain observed froin gages 1 and 2, when the external hydi-ostatic pres- sure was applied directly on the pipe, are given in Figs. 5 and 6. Corre- spondmg curves when the external pressure was applied on the model are also plotted in these figures. The lines pa.ss through the origin because the bridge was initially balanced at zero pressure with the help of blaiicing resistances in the multichannel switching unit, and in each case the zero reading was checked after the experiment, when the pressure was brought back to zero. The deviation of the experimental points from the straight hnes for low values of pressure is probably due to the nonlinearity in the pressure gages, which are designed to be used normally at much higher pressures. It may be noted that, in the case of Lame's equations for thick-walled cylinders, the sum of the radial and circumferential stresses is a con- stant at every point in the cross section, so that the deformation of all elements in the direction of the axis of the cylmder is the same and cross sections of the cylinder remain plane after deformation. Hence, in spite of the considerable length of the cyhnder, the experunental arrangement gives the verification for plane stress conditions. PRESSURE AT IMTERFACE IX ARMORLESS OCEAN CABLE 967 5000 f O z 1^ / u^OOO cc = 3500 10 CALIBRATION FOR STEEL CYLINDER^ / Z' / / y y 0. y / y ^ z 3 / / ^ y z / r / /concentric STEEL AND POLYETHYLENE CYLINDERS PRESSURE 8 i 4' y ^T. / /y y y 400 600 eoo STRAIN, INCH PER INCH 1000 I2O0XI0'' Fig. 5 — Pressure vs, strain from giige 1. The pressure iit the interface is obtained hy druwinfj; a vertical Hne from the observed vahie of strain to the caliliration line for the steel cylinder alone and reading the eorrespondmg pressure. Numerical calcu- lations from Figs. 5 and 6 show that the pressure at the interface, in the case of plane stress is 1 .3 times the external pressure. 5000 z / / a. g3500 in CALIBRATION FOR STEEL CYLINDER / /I / ¥^ Q. / -£y / z D 2 2000 z UJ 1500 a: ./ V y 1 / / /concentric steel AND '^ POLYETHYLENE CYLINDERS / ^ •■^ 0. 9 /- r i^ i 200 400 600 800 STRAIN, INCH PER INCH 1200 X 10^ Fig. — Pressure vs. strain from gage 2. 968 THE BICLL SYSTEM TECHNICAL JOURNAL, JULY 1960 ,-- — STEEL - — POLYETHYLENE Fig.. 7 — Cross section of the model. IV. THEORETICAL DERIVATION OF PRESSURE AT THE INTERFACE The differential cqimtion of equilibrium,^ in terms of radial displace- ment, for a thick-walled cylinder subjected to external and intermal pressures is du 1 du ^* _ r, dr"^ r dr t' The general solution of the above equation is (1) r (2) Substituting (2) in the expressions for Hooke's law fur plane stress, one obtams for radial stress, a, 1 -m4 Cl(l + ;U) — C2 (1 - 1 -m) 1 r= J' (3) where Ci and Ca are arbitrary constants and E and m are Young's modulus and Poisson's ratio respectively. For the model shown in Fig. 7, the following boundary conditions must be satisfied; (4) (5) PKESSURE AT INTERFACE IN ARMORLESS OCEAN CABLE 969 C,\r=.= 0, (6) Up r=6 ^ Ws T=b • (7) In the above equations subscripts p and s stand for polyethylene and steel. Substituting (2) and (3) in (4) through (7), one gets: ^[.(i + .)-.(Lz^')] ^-A^[.a + .)-c.(L^-)]. K r 1 - Mp^ L Ci(l + Up) — G2 (1 - 1 - Mp) 1 C2 J = ?Jo, ,; Cad + mJ - Ci -T. — ■ = 0, Cih -h ~ = Czb + r* . (8) (9) (10) (11) The pressure p at the interface is obtained by solving for the constants from the foregoing equations and evaluating (Tp I r=b = p. Equation (12) gives the pressure at the interface for a plane stress solution. P = VoS (b' -c)\^(l -,.p)(h' -a') - (1 - M.)&' - (1 + M.)«' (6^ -c')[(l - mJ&' + (1 + M.)a1 -^(b' - a') [(1 - ^p)b' + (1 + ^ip)c] hp + . (12) To obtain the plane strain solution, the constants E and n in (12) are replaced by E* and m*, with appropriate subscripts given by the follow- ing; E* = M* = B It , 1 -m' (13) (14) 970 THE BELL SYSTEM TECHNICAL JOUHNAL, JULY ] 9G0 V. NUMERICAL EXAMPLE FOB THE MODEL The values for the various constants are : a = 1.088 inches, h — 1.254 inches, c = 4.621 inches, i:. - 29 X 10' psi, 7?^ = 19 X 10' psi, Us = 0.3, M;, = 0.5. Substitution of above values in (12) gives for tlie plane stress solution V= 1.3(po). (15) Replacing E and n by (13) and (14) and substituting in (12), one obtains for the plane strain solution V = l-0(p«). (16) It may be noticed from (12) that the pressure at the interface ap- pears to depend on the values of Ep and Hp . However, examination reveals that (12) is relatively msensitive to Ep , for the range of known values for low and intermediate density polyethylene. Since it is difficult to determine tip , one may use the previously known values of bulk modulus, Kj, , for polyethylene and obtaui Hp from Refs. 3 and 4 give the value of Kp ^ 0.5 X 10^ psi. Ref. 5 gives for E,, , for low and intermediate density, the range of values from 19 X 10^ to 55 X 10' psi. With Kp = 0.5 X lO' psi and Ep - 19 X lO' or 55 X lO' psi, (17) gives Mp to be very nearly equal to 0.5. It may be noted that the analysis for concentric cylinders given in this paper agrees with experimental results when /Zp is very nearly equal to 0.5. VI. CONCLUSION Experiments indicate that an analysis based on equations of the two- dimensional theory of elasticity for the pressure at the mterface of polyethylene outer and steel inner concentric cylinders, under plane PRESSURE AT INTERFACE IN ARMORLESS OCEAN CABLE 071 stress conditions subjected to a uniform pressure on the curved surface, is valid even for polyethylene, which is a material with nonlinear stress- strain relationship. The agreement between the theoretically calculated and experimentally determined values is very good. It is also shown that the value of Poisson's ratio for low and intermediate density poly- ethylene, for an elastic analy.sis, should be very nearly 0.5. The experi- mental verification of the plane stress .solution given in the present paper confirms this value. An experimental verification of the plane strain solution involves a number of difficulties in satisfying the condition that the axial strain be zero; it has not been attempted in the present paper. However, one may assume that the plane strain solution is also valid, because the plane stress solution obtained by using the same two-dimensional theory of elasticity has been experimentally verified. The plane strain solution gives for the numerical example the result that the pressure at the inter- face is the same as external pressure. The cable is actually under a uniform pressure in three dimensions. The two-dimensional plane strain solution may be extended to a three- dimensional case of uniform pressure by a superposition of an axial loading of intensity po and satisfying the appropriate boundary and compatibility conditions. However, since the value of Poisson's ratio for polyethylene is 0.5 and the area of polyethylene is much greater in comparison to the area of the steel, the pre.ssure at the interface of the inner conductor and the polyethylene dielectric, as given by the plane strain solution, would not be much altered. Hence, the pressure at the interface of the mner conductor and the dielectric of the armorless ocean cable is equal to or at most slightly greater than the external pressure. It may be remarked that a new experimental method employing re- sistance wire strain gages has been developed to measure interface pres- sures accurately. The method devised in this paper could be used in other applications to mea.sure interface pres.sures. VII. LIST OF SYMBOLS a = inner radius of steel cylinder (inches). h = outer radius of steel cylinder (inches), c = outer radius of polyethylene cylinder (inches). Kp = bulk modulus for polyethylene (psi). p = pres.sure at the interface (psi). ^,(1 = external hydrostatic pres^;urc (psi). u ^ radial displacement (inches). r = radial distance (inches). 972 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1900 E = Young's modulus (psi). Ep = Young's modulus for polyethylene (psi). E, = Young's modulus for steel (psi). n = Poisson's ratio. fip = Poisson's ratio for polyethylene. fig = Poisson's ratio for steel. a = radial stress (psi). tTp = radial stress in polyethylene (psi). a, = radial stress m steel (psi). REFERENCES 1. Timoflhenko, S., Strength of Materials, Part II, D. Van Nostrand Co., New York, 1956, pp. 205-214. 2. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book Co., New York, 1951, pp. 34; 54-60. 3. Bridgman, P. W., Proc. Amer. Acad. Arts & Sei., 76, 1948, p. 71. 4. Weir, C. E., National Bureau of Standards, Research Paper 2192, 1951. 5. Modern Plastics Encyclopedia, Plastics Properties Chart, Breskin Publications, New York, 1959.